# Nash equilibrium two players in a joint effortful project

I have a problem where there are two agents in a joint project, Each agent $$i$$ puts in effort $$x_i$$ $$(0 \leq x_i \leq 1)$$ which cost each $$c(x_i)= x_i^2$$. The outcome of the project is $$f(x_1, x_2)= 3 x_1 x_2$$

which is split equally between both, regardless of their effort levels. I am asked to formulate de situation as a normal form game and find the NE.

What I have done is that I have assumed some values for the efforts to evaluate the game and I have also maximized each player's payoff individually to get best responses and make them intersect to get to the NE.

1. In this illustration, NE are $$(0,0)$$; $$(0.25, 0.25)$$ and $$(0.5, 0.5)$$

2. $$\max f(x_1, x_2)-c_1$$ with respect to $$x_1$$ ; yields : $$x_1 = (3/4)x_2$$ and $$\max f(x_1, x_2)-c_2$$ with respect to $$x_2$$ ; yields : $$x2 = (3/4)x_1$$ Both maximization problems intersect when $$x_1 = 0 = x_2$$

Anyhow, I don't think this is the right way to go...